Localization of Discrete Spectrum of Multiparticle Schrödinger Operators

1985 ◽  
Vol 40 (10) ◽  
pp. 1052-1058 ◽  
Author(s):  
Heinz K. H. Siedentop
Keyword(s):  
Discrete Spectrum ◽  
Lower Bounds ◽  
Upper Bound ◽  
Schrödinger Operators ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators

An upper bound on the dimension of eigenspaces of multiparticle Schrödinger operators is given. Its relation to upper and lower bounds on the eigenvalues is discussed.

scholarly journals Spectral Analysis of Schrödinger Operators with Unusual Semiclassical Behavior

10.14311/1801 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Pavel Exner ◽  
Diana Barseghyan
Keyword(s):  
Spectral Analysis ◽  
Phase Space ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Schrödinger Operators ◽  
Critical Value ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators ◽  
Two Dimensional ◽  
Eigenvalue Bounds

In this paper we discuss several examples of Schrödinger operators describing a particle confined to a region with thin cusp-shaped ‘channels’, given either by a potential or by a Dirichlet boundary; we focus on cases when the allowed phase space is infinite but the operator still has a discrete spectrum. First we analyze two-dimensional operators with the potential |xy|p - ?(x2 + y2)p/(p+2)where p?1 and ??0. We show that there is a critical value of ? such that the spectrum for ??crit it is unbounded from below. In the subcriticalcase we prove upper and lower bounds for the eigenvalue sums. The second part of work is devoted toestimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions havinginfinite cusps which are geometrically nontrivial being either curved or twisted; we are going to showhow these geometric properties enter the eigenvalue bounds.

Localization of Eigenvalues by a Modification of Müller′s Variational Principle in Some Exactly Solvable Cases

1983 ◽  
Vol 38 (5) ◽  
pp. 493-496 ◽  
Author(s):  
Heinz K. H. Siedentop
Keyword(s):  
Variational Principle ◽  
Lower Bounds ◽  
Schrödinger Operators ◽  
Three Dimensional ◽  
Finite Depth ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators ◽  
Potential Well ◽  
Rough Approximation ◽  
Exactly Solvable

Upper and lower bounds on the eigenvalues of Schrödinger operators with simple one and a simple three dimensional potential (well of finite depth, spherical δ-potential) are given by means of a modification of Müller′s variational principle. The estimates, comparing them with the exact eigenvalues, show a localization of the eigenvalues even in a rough approximation for the trial operator.

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals Discrete spectrum of many body Schrödinger operators with non-constant magnetic fields II

1997 ◽  
Vol 145 ◽  
pp. 69-98
Author(s):  
Tetsuya Hattori
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Discrete Spectrum ◽  
Bound States ◽  
Atomic System ◽  
Energy Levels ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Many Body ◽  
Discrete Energy

This paper is continuation from [10], in which we studied the discrete spectrum of atomic Hamiltonians with non-constant magnetic fields and, more precisely, we showed that any atomic system has only finitely many bound states, corresponding to the discrete energy levels, in a suitable magnetic field. In this paper we show another phenomenon in non-constant magnetic fields that any atomic system has infinitely many bound states in a suitable magnetic field.

scholarly journals When is non-trivial estimation possible for graphons and stochastic block models?‡

2017 ◽  
Vol 7 (2) ◽  
pp. 169-181
Author(s):  
Audra McMillan ◽  
Adam Smith
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Random Networks ◽  
Block Models

Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

scholarly journals Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

2019 ◽  
Author(s):  
Wencai Liu
Keyword(s):  
Upper Bound ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Embedded Eigenvalues ◽  
Finite Set ◽  
Discrete Schrödinger Operators ◽  
Sign Type ◽  
Free Operator ◽  
Set Of Points ◽  
Rational Type

Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$ and $H_0$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } n|V(n)|=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(-2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(-2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $|V(n)|\leq \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

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